Connecting two semicontinuous processes with independent increments

Abstract

We consider a stopped process X_t ⁰ in the phase space E⁰=(−∞, +∞)/{0} such that X_t ⁰=X_t ¹ if X_t ⁰ > 0 and X_t ⁰=X_t ² if X_t ⁰ < 0, where X_t ^j, j=1,2, are nonstopped stochastically continuous Markov processes with independent increments and with only negative jumps. We prove that there exists an extension of X_t ⁰ into a homogeneous, stochastically continuous, and strong Markov Feller process X_t in the phase space (−∞; +∞) and that the extension can be characterized by a measure N(dy) and three constants b, c₁ c₂.